On Monday at the U.S. Open, Serena Williams strolled to just the sort of routine victory she has recorded with such regularity over her long and illustrious career. This one, though, was just a little bit more special.

A 6-2 6-3 triumph over Yaroslava Shvedova gave Williams her 308th victory in a Grand Slam, eclipsing the previous record held by Roger Federer.

“I think it’s really exciting, she said afterward. “Winning 308 matches in general is pretty awesome, for that to be in a grand slam is pretty cool. It’s a huge number. I think it’s very significant actually, it’s something that really talks about the length of my career – I've been playing for a really long time – but also the consistency out there and that’s something that I'm really proud of.

“I think it just comes from a different place of the just love and really enjoying it. I definitely never thought I would be playing still. Now I don't really see when I'm going to stop because I'm just enjoying these moments out here, getting to break records that I didn't even know existed or I didn't even know was possible.”

Perhaps even more significantly, the win put the 34-year-old into the quarterfinals of the U.S. Open and just three victories away from an even more significant record. If Williams holds the trophy aloft at Flushing Meadows on Saturday she will eclipse Steffi Graf’s Open-era record of 22 Grand Slam titles.

Thus far in New York there has been little sign that anyone will be able to stop her. The shoulder injury that cast doubt over her prospects coming into the final major of the tennis season has been in little evidence. Indeed, her serve has been as dominant as ever and she has faced just one break point through her first four matches, all straight-sets wins.

There has been little indication, either, of the pressure brought on by chasing such a momentous achievement weighing on her to the extent it did 12 months ago in New York when she suffered a stunning semifinal upset at the hands of Roberta Vinci.

Serena Williams Serena Williams has shown little sign of physical or mental stress on her route through to the U.S. Open quarterfinals. Photo: Getty Images

She has, though, yet to face a seeded player and the level of competition will increase significantly on Wednesday. In the evening session on Arthur Ashe Stadium, Williams will take on fifth seed Simona Halep, one of the form players this summer. Indeed since the French Open in June the Romanian has only been beaten by world No. 2 Angelique Kerber and has claimed titles in Bucharest and Montreal.

Still, Williams has dominated the pair’s rivalry, with Halep’s only victory in their eight matches coming in the round-robin phase of the 2014 WTA Finals. And Williams almost immediately gained emphatic revenge, brushing Halep aside for the loss of just three games in the tournament final. Unsurprisingly, Williams gave no indication that she was paying any particular attention to the threat posed by the 24-year-old.

“I just think with everyone I play, they play a step up and above their ranking,” she said. “I think that's the beauty, one of the reasons I'm able to hang. Everyone I'm playing is playing like they're No. 1.

To me it doesn't really matter who I play because I have to expect they're going to play the match of their life. That's how I go into these matches now.”

Should Williams get through that challenge she will be spared the emotional toll, and significant physical test, of taking on her sister. Venus Williams lost in a three-set thriller to Karolina Pliskova in the fourth round and the Czech will now take on teenage sensation Ana Konjuh in the quarterfinals for the right to meet Serena or Halep.

Williams would be an overwhelming favorite against either woman. Indeed many have long since been anticipating a third Grand Slam final this year between Williams and Kerber. The German grabbed her first Grand Slam title by beating Williams at the Australian Open in January before Williams got her revenge at Wimbledon to equal Graf’s mark of 22 major titles.

Number 23 could now be just days away, leaving Margaret Court’s all-time record of 24 Grand Slam titles, and confirmation, as if it were needed, as the greatest female player of all time firmly in her sights.